It is quite well explained here -- https://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/
Although, I don't truly understand, I see it as an attempt to calculate the compound interest curve; and natural logarithms as reverse calculating it.
Feel that this website is about right -- https://betterexplained.com/archives/ (and there are computer science problems, too.)
Though, another approach is to start from problems (thought or real).
BTW, my favorite in computer thingy might be,
float Q_rsqrt( float number ) { long i; float x2, y; const float threehalfs = 1.5F; x2 = number * 0.5F; y = number; i = * ( long * ) &y; // evil floating point bit level hacking i = 0x5f3759df - ( i >> 1 ); // what the fuck? y = * ( float * ) &i; y = y * ( threehalfs - ( x2 * y * y ) ); // 1st iteration // y = y * ( threehalfs - ( x2 * y * y ) ); // 2nd iteration, this can be removed return y; } 
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E is one of my favourite number. 😍